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<h2><span class="refentrytitle">Struct template peaks_over_threshold_impl</span></h2>
<p>boost::accumulators::impl::peaks_over_threshold_impl — Peaks over Threshold Method for Quantile and Tail Mean Estimation. </p>
</div>
<h2 xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" class="refsynopsisdiv-title">Synopsis</h2>
<div xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" class="refsynopsisdiv"><pre class="synopsis"><span class="comment">// In header: &lt;<a class="link" href="../../../accumulators/reference.html#header.boost.accumulators.statistics.peaks_over_threshold_hpp" title="Header &lt;boost/accumulators/statistics/peaks_over_threshold.hpp&gt;">boost/accumulators/statistics/peaks_over_threshold.hpp</a>&gt;

</span><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Sample<span class="special">,</span> <span class="keyword">typename</span> LeftRight<span class="special">&gt;</span> 
<span class="keyword">struct</span> <a class="link" href="peaks_over_threshold_impl.html" title="Struct template peaks_over_threshold_impl">peaks_over_threshold_impl</a> <span class="special">:</span> <span class="keyword">public</span> <span class="identifier">accumulator_base</span> <span class="special">{</span>
  <span class="comment">// types</span>
  <span class="keyword">typedef</span> <span class="identifier">numeric</span><span class="special">::</span><span class="identifier">functional</span><span class="special">::</span><span class="identifier">fdiv</span><span class="special">&lt;</span> <span class="identifier">Sample</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="special">&gt;</span><span class="special">::</span><span class="identifier">result_type</span> <a name="boost.accumulators.impl.peaks_over_threshold_impl.float_type"></a><span class="identifier">float_type</span><span class="special">;</span> 
  <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">tuple</span><span class="special">&lt;</span> <span class="identifier">float_type</span><span class="special">,</span> <span class="identifier">float_type</span><span class="special">,</span> <span class="identifier">float_type</span> <span class="special">&gt;</span>            <a name="boost.accumulators.impl.peaks_over_threshold_impl.result_type"></a><span class="identifier">result_type</span><span class="special">;</span>
  <span class="keyword">typedef</span> <span class="identifier">mpl</span><span class="special">::</span><span class="identifier">int_</span><span class="special">&lt;</span> <span class="identifier">is_same</span><span class="special">&lt;</span> <span class="identifier">LeftRight</span><span class="special">,</span> <a class="link" href="../left.html" title="Struct left">left</a> <span class="special">&gt;</span><span class="special">::</span><span class="identifier">value</span> <span class="special">?</span> <span class="special">-</span><span class="number">1</span> <span class="special">:</span><span class="number">1</span> <span class="special">&gt;</span>        <a name="boost.accumulators.impl.peaks_over_threshold_impl.sign"></a><span class="identifier">sign</span><span class="special">;</span>       

  <span class="comment">// <a class="link" href="peaks_over_threshold_impl.html#boost.accumulators.impl.peaks_over_threshold_implconstruct-copy-destruct">construct/copy/destruct</a></span>
  <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <a class="link" href="peaks_over_threshold_impl.html#id-1_3_2_6_3_19_1_1_10_1_9-bb"><span class="identifier">peaks_over_threshold_impl</span></a><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="special">)</span><span class="special">;</span>

  <span class="comment">// <a class="link" href="peaks_over_threshold_impl.html#id-1_3_2_6_3_19_1_1_10_1_8-bb">public member functions</a></span>
  <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="keyword">void</span> <a class="link" href="peaks_over_threshold_impl.html#id-1_3_2_6_3_19_1_1_10_1_8_1-bb"><span class="keyword">operator</span><span class="special">(</span><span class="special">)</span></a><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="special">)</span><span class="special">;</span>
  <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="identifier">result_type</span> <a class="link" href="peaks_over_threshold_impl.html#id-1_3_2_6_3_19_1_1_10_1_8_2-bb"><span class="identifier">result</span></a><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Archive<span class="special">&gt;</span> <span class="keyword">void</span> <a class="link" href="peaks_over_threshold_impl.html#id-1_3_2_6_3_19_1_1_10_1_8_3-bb"><span class="identifier">serialize</span></a><span class="special">(</span><span class="identifier">Archive</span> <span class="special">&amp;</span><span class="special">,</span> <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="keyword">int</span><span class="special">)</span><span class="special">;</span>
<span class="special">}</span><span class="special">;</span></pre></div>
<div class="refsect1">
<a name="id-1.3.2.6.3.18.4.4"></a><h2>Description</h2>
<p>According to the theorem of Pickands-Balkema-de Haan, the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_90.png"></span> of the excesses <span class="inlinemediaobject"><img src="../../../images/accumulators//form_91.png"></span> over some sufficiently high threshold <span class="inlinemediaobject"><img src="../../../images/accumulators//form_92.png"></span> of a distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_93.png"></span> may be approximated by a generalized Pareto distribution </p>
<div class="equation">
<a name="id-1.3.2.6.3.18.4.4.2.5"></a><p class="title"><b>Equation 1.1. </b></p>
<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_94.png" align="middle"></div></div>
</div>
<p><br class="equation-break"> with suitable parameters <span class="inlinemediaobject"><img src="../../../images/accumulators//form_88.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators//form_95.png"></span> that can be estimated, e.g., with the method of moments, cf. Hosking and Wallis (1987), </p>
<div class="equation">
<a name="id-1.3.2.6.3.18.4.4.2.8"></a><p class="title"><b>Equation 1.2. </b></p>
<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_96.png" align="middle"></div></div>
</div>
<p><br class="equation-break"> <span class="inlinemediaobject"><img src="../../../images/accumulators//form_97.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators//form_98.png"></span> being the empirical mean and variance of the samples over the threshold <span class="inlinemediaobject"><img src="../../../images/accumulators//form_92.png"></span>. Equivalently, the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_99.png"></span> of the exceedances <span class="inlinemediaobject"><img src="../../../images/accumulators//form_100.png"></span> can be approximated by <span class="inlinemediaobject"><img src="../../../images/accumulators//form_101.png"></span>. Since for <span class="inlinemediaobject"><img src="../../../images/accumulators//form_102.png"></span> the distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_93.png"></span> can be written as </p>
<div class="equation">
<a name="id-1.3.2.6.3.18.4.4.2.17"></a><p class="title"><b>Equation 1.3. </b></p>
<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_103.png" align="middle"></div></div>
</div>
<p><br class="equation-break"> and the probability <span class="inlinemediaobject"><img src="../../../images/accumulators//form_104.png"></span> can be approximated by the empirical distribution function <span class="inlinemediaobject"><img src="../../../images/accumulators//form_105.png"></span> evaluated at <span class="inlinemediaobject"><img src="../../../images/accumulators//form_92.png"></span>, an estimator of <span class="inlinemediaobject"><img src="../../../images/accumulators//form_93.png"></span> is given by </p>
<div class="equation">
<a name="id-1.3.2.6.3.18.4.4.2.22"></a><p class="title"><b>Equation 1.4. </b></p>
<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_106.png" align="middle"></div></div>
</div>
<p><br class="equation-break"> It can be shown that <span class="inlinemediaobject"><img src="../../../images/accumulators//form_107.png"></span> is a generalized Pareto distribution <span class="inlinemediaobject"><img src="../../../images/accumulators//form_108.png"></span> with <span class="inlinemediaobject"><img src="../../../images/accumulators//form_109.png"></span> and <span class="inlinemediaobject"><img src="../../../images/accumulators//form_110.png"></span>. By inverting <span class="inlinemediaobject"><img src="../../../images/accumulators//form_107.png"></span>, one obtains an estimator for the <span class="inlinemediaobject"><img src="../../../images/accumulators//form_20.png"></span>-quantile, </p>
<div class="equation">
<a name="id-1.3.2.6.3.18.4.4.2.29"></a><p class="title"><b>Equation 1.5. </b></p>
<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_111.png" align="middle"></div></div>
</div>
<p><br class="equation-break"> and similarly an estimator for the (coherent) tail mean, </p>
<div class="equation">
<a name="id-1.3.2.6.3.18.4.4.2.30"></a><p class="title"><b>Equation 1.6. </b></p>
<div class="equation-contents"><div class="mediaobject" align="center"><img src="../../../images/accumulators//form_85.png" align="middle"></div></div>
</div>
<p><br class="equation-break"> cf. McNeil and Frey (2000).</p>
<p>Note that in case extreme values of the left tail are fitted, the distribution is mirrored with respect to the <span class="inlinemediaobject"><img src="../../../images/accumulators//form_112.png"></span> axis such that the left tail can be treated as a right tail. The computed fit parameters thus define the Pareto distribution that fits the mirrored left tail. When quantities like a quantile or a tail mean are computed using the fit parameters obtained from the mirrored data, the result is mirrored back, yielding the correct result.</p>
<p>For further details, see</p>
<p>J. R. M. Hosking and J. R. Wallis, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, Volume 29, 1987, p. 339-349</p>
<p>A. J. McNeil and R. Frey, Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: an Extreme Value Approach, Journal of Empirical Finance, Volume 7, 2000, p. 271-300</p>
<p>
</p>
<div class="refsect2">
<a name="id-1.3.2.6.3.18.4.4.8"></a><h3>
<a name="boost.accumulators.impl.peaks_over_threshold_implconstruct-copy-destruct"></a><code class="computeroutput">peaks_over_threshold_impl</code> 
        public
       construct/copy/destruct</h3>
<div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem"><pre class="literallayout"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <a name="id-1_3_2_6_3_19_1_1_10_1_9-bb"></a><span class="identifier">peaks_over_threshold_impl</span><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</span> args<span class="special">)</span><span class="special">;</span></pre></li></ol></div>
</div>
<div class="refsect2">
<a name="id-1.3.2.6.3.18.4.4.9"></a><h3>
<a name="id-1_3_2_6_3_19_1_1_10_1_8-bb"></a><code class="computeroutput">peaks_over_threshold_impl</code> public member functions</h3>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem"><pre class="literallayout"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="keyword">void</span> <a name="id-1_3_2_6_3_19_1_1_10_1_8_1-bb"></a><span class="keyword">operator</span><span class="special">(</span><span class="special">)</span><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</span> args<span class="special">)</span><span class="special">;</span></pre></li>
<li class="listitem"><pre class="literallayout"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Args<span class="special">&gt;</span> <span class="identifier">result_type</span> <a name="id-1_3_2_6_3_19_1_1_10_1_8_2-bb"></a><span class="identifier">result</span><span class="special">(</span><span class="identifier">Args</span> <span class="keyword">const</span> <span class="special">&amp;</span> args<span class="special">)</span> <span class="keyword">const</span><span class="special">;</span></pre></li>
<li class="listitem"><pre class="literallayout"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> Archive<span class="special">&gt;</span> 
  <span class="keyword">void</span> <a name="id-1_3_2_6_3_19_1_1_10_1_8_3-bb"></a><span class="identifier">serialize</span><span class="special">(</span><span class="identifier">Archive</span> <span class="special">&amp;</span> ar<span class="special">,</span> <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="keyword">int</span> file_version<span class="special">)</span><span class="special">;</span></pre></li>
</ol></div>
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